The first time she was pregnant it didn't enter Karen Rodger's head that she might be having twins. Likewise, the second time. The third time she thought she had miscarried and wasn't having one child, let alone two.

Today, Karen and her husband, Colin, have defied odds of 88,000 to one to produce three sets of non-identical twins. Some mothers, having given birth to even one, let alone two sets of multiples, might have yearned for the relative simplicity of a single baby – but not Karen. Throughout the afternoon I spend with her at the family home overlooking the river Clyde outside Glasgow, her smile never wanes. The newest arrivals are just a few days old: "I keep waiting for the hormonal dip," she says. "But it never comes. I'm elated."

A third set of twins, it seems, never had a more welcome landing, even though Karen and Colin had given away all their baby equipment, times two, after the middle set of twins, who are now 12. "We'd never have imagined it could happen again," says Karen. All she did know is that, when she hit 40 a year ago, a little voice inside her whispered that she hadn't quite finished with childbearing. Could it have been because as the mother of four boys – Lewis and Kyle are 14, Finn and Jude are 12 – she was yearning for a daughter? Definitely not, she says. "I didn't just want a daughter, I wanted another baby. I knew it wasn't very likely to be a daughter – after four of them, I thought we could probably only make boys."

Her most recent pregnancy started badly: no sooner had she discovered that she was expecting than she had what appeared to be a miscarriage. Her doctor suggested she should go for a scan, but Colin had work commitments. "He felt awful about not being able to come with me, but I said I'd be fine: whatever happened, I'd cope," says Karen.

"There I was, in the middle of this meeting," says Colin, "when I got a text from Karen. I think I'd texted something like: 'Is everything OK?' And she texted back: 'Yes, and it looks like it's twins.' I said something like 'that's not funny', excused myself from the meeting and went outside to phone her. And she said it was really true, that the scan operator thought that, though it was early days, it was going to be two babies again.

"Then I had to go back into the meeting and sit there, in total shock, without being able to tell anyone. Although part of me was actually laughing inside: because I was thinking to myself, why did we ever think it would be any different?"

Karen and Colin, who were both born and raised near Glasgow, met 20 years ago when she was a dancer (she now lectures in dance) and he was managing a band. Her maternal great-grandmother had had twins, but they had no inkling of what lay ahead when she got pregnant with Lewis and Kyle in 1998. The discovery that she was carrying twins explained why Karen had been feeling so sick – hyperemesis gravidarum, or extreme morning sickness, is more common in women carrying multiples. All the same, when she got pregnant again two years later and started to feel terrible, she didn't think for a second that it would be for the same reason. "When I went for my scan, it was the same operator as the first time round, and when she said it was another set I said 'very funny'. But she said it's not a joke. Colin was in complete denial; he just couldn't take it in."

But the third time round really did confound the odds: although the Guinness Book of Records documents an 18th-century Russian woman who gave birth to 16 sets of twins, the chances of a third twin conception are very unlikely. "And, of course, no one even imagined I'd be having another baby at all, let alone twins – the older boys were so big, and people would have assumed childbearing was behind me," says Karen.

The first people they told, once the news was confirmed, were the boys. "They brought us all down to the sitting room and told us to sit down," says Kyle. "And I thought, this is going to be a big row about how untidy the house is. I was sitting there expecting to get shouted at, and suddenly Dad said: 'Mum's pregnant, and it's two.' We all sat there in total silence: none of us knew what to say. And then eventually I said: 'Well, I'm definitely never going to get my own bedroom now.'"

Karen and Colin decided not to find out the sex of the babies she was carrying but she was certain it was another pair of boys, which in turn convinced everyone else. As with the previous sets of twins, the new arrivals were born by caesarean. "The obstetrician lifted Rowan out and held her up for us to see," says Colin. "And I said: 'There are some bits missing.' And she said, 'No, it's a girl,'" says Colin. "I was absolutely stunned. Then two minutes later they held Isla up and I could see straight away she was another girl. Karen and I were in shock for at least the first 15 minutes. It's a lot to take in after four boys."

Karen Rodger with new arrivals Isla
(left) and Rowan, her third set of twins. Photograph: Tom Finnie for the Guardian Tom Finnie/Guardian

Today, the hallway of the Rodgers' home is crammed with football kit and muddy boots, but the sitting room is a sea of pink, the room festooned in flowers, balloons and cards. "I wasn't sure how I was going to cope with all the pink – it's very different after 14 years of boys' stuff. I'm getting my head round the idea that they'll be borrowing my makeup and sparkly shoes," says Karen. Lewis, meanwhile, is cradling Rowan and can't take his eyes off her. "I'd never held a tiny baby before, but I'm getting used to it," he says.

Not surprisingly, Karen and Colin are taking the arrival of the newest twins in their stride: they are old hands, after all. "We know how exhausting the early weeks are when you've got two babies, but we also know we'll come through it and things will move on," says Colin. "We've got a huge amount of help – both sets of grandparents are very supportive, and I'd even say that we wouldn't be in this position, with all these children, if it hadn't been for them being right behind us."

As we're talking, Karen's mum is tackling some of the mountain of ironing (they have the biggest box of soap powder I've ever seen – a family with four football-playing boys and two babies would surely be a dream setting for a detergent commercial) and Colin says this is the first day since the babies were born when his mum hasn't been over as well.

But it's the overnight arrangements that are often most difficult for parents of newborn multiples; so how do the Rodgers cope? "Karen tends to go to bed early and I do the 11pm feed with her expressed milk in bottles," says Colin. "Then when the babies wake in the night either Karen will feed both of them or she'll breastfeed one and I'll bottlefeed the other. We change their nappies, Karen expresses some more milk, and then we all go back to sleep for a few hours.

"If I've got an important day at work I might sleep somewhere else. Karen has lots of friends in to help during the day so she tries to catch up on her sleep then." Karen, meanwhile, says that she's realised through her long years of caring for twins that the practical solution isn't always the best one: sometimes, even though it takes longer, she has learned to concentrate on them one at a time. "The temptation is that when one wants to be fed, you wake the other so you can feed them together and save time. But I see feeding time as bonding time, so I treasure the one-on-one experience when there's an opportunity."

One thing that's very obvious in the Rodger household – and both parents feel very strongly about this – is that they're not raising three sets of twins, they're raising six children. "Our children are all very different," says Colin. "And in fact the boys are more like their brother who isn't their twin than their brother who is: Kyle and his younger brother Jude tend to be similar in that they're more compliant, they don't like to make a fuss, whereas Finn and Lewis are both more hardcore and resilient." Karen says she has never dressed her twins alike or made assumptions that because one likes something, the other will too. "They've got their own interests, and as they've grown up they've developed their own friendships," she says. And indeed, the house today is full to bursting with a procession of teenage boys. Some of them are even being allowed tentative holds of the newborns, overseen by their protective big brothers. The Rodger boys are reliving the moments when they found out about the birth. "Dad sent me a picture of these two tiny baby girls on my mobile, but I still didn't twig," says Kyle. "I texted back: 'Why did you send me that picture?' And he said: 'They're your sisters.'" Finn and Jude were called out of class to the staff room so their dad could tell them on the phone. "They were holding the phone between them and Colin said, 'we've got two little girls' and the boys said 'girls!' and the room erupted behind them," says Karen.

But there are some daunting elements to raising six children, not least money. "Having four costs a fortune, so six is going to go through the roof," says Colin. "We've got to get a new car, and it's going to have to be a big one. And we've told the boys we'll extend the house, because the older two really do want their own bedrooms." Plus they've had to invest in a double pram, two cots and two car seats. As 12-year-old Finn puts it: "It's like the ark round here. Everything is two by two, by two by two – and now it's two by two again."

The odds of having twins

• Non-identical twins are about twice as common as identical twins. The incidence of non-identicals in the UK is one in 112 births, compared with one in 227 births for identicals.

• Identical twins are down to chance: one egg is fertilised, but it then divides. Fraternal twins happen when a woman produces two eggs, and both are fertilised. So with identical twins the DNA makeup is exactly the same in each twin; with fraternal twins, the DNA is no more similar than in any two siblings.

• A history of non-identical twins on the mother's side increases her chance of twins; and a woman who already has twins has four times the normal chance of having them again in a subsequent pregnancy.

• Increased maternal age makes twins more likely, because older ovaries are more likely to release two eggs in the same month.

• The odds of having three sets of non-identical twins is about one in 88,000, but the chance of having three sets of identical twins is one in 11.7m.